Scheduling and Sorting Algorithms for Robotic Warehousing Systems Master Thesis Daniel Graf July 18, 2015 Advisors: Prof. Dr. P. Widmayer and K. B¨ohmov´a Department of Computer Science, ETH Z¨urich Abstract Automated storage and retrieval systems are used in many warehouses for time and space efficient stocking and order fulfilment. In this thesis, we study such automated systems in the context of storing bicycles at a train station, where commuters bring their bicycles in the morning and reclaim them in the evening. The commuters place their bicycles in a box at a door, where a robot then moves it underground for safe storage. In the first part of the thesis, we investigate how the robot best assigns boxes to the underground storage slots. The goal is to find a schedule which allows the robot to meet all the customers at the door without causing any waiting time. For a single door, we devise an efficient al- gorithm that finds such an optimal schedule under certain conditions. If there is a second door, or arriving and departing customers are heav- ily interleaved, we show that it is -complete for the robot to find a wait-free schedule even if it knowsNP all the requests in advance. In the second part of the thesis, we focus on the task of rearranging the stored boxes in between customer interactions. We phrase the task as a compelling physical sorting problem: on every vertex of a graph G, we place a box. The n vertices and n boxes are each numbered from 1 to n, and initially shuffled according to a permutation p. A single robot is given the task to sort these boxes. In every step, the robot can walk along an edge of the graph and can carry at most one box at a time. At a vertex, it may swap the box placed there with the box it is carrying. How many steps does the robot need to sort all the boxes? We present efficient algorithms that construct such a shortest sorting walk if G is a path, a tree or a cycle, and we show that the problem is -complete for planar graphs. NP i Acknowledgements First and foremost, I would like to express my gratitude to Katerinaˇ Bohmov¨ a´ and Peter Widmayer for advising me during this thesis. I enjoyed the interesting and helpful discussions in our weekly meetings immensely. They encouraged me to keep exploring this fascinating topic throughout these months by always asking the right questions and contributing countless insights. I would like to thank Armin Wyttenbach of Bikeloft GmbH and Matthias Wilhelm of Nussbaum Technologies GmbH for taking the time to ex- plain their exciting automated bicycle storage systems to me. Special thanks go to Filip Pavetic´ for pointing me to the Croatian task Ku- tije and to Sandro Feuz for always listening to and dissecting my half- baked ideas and proofs. I also got valuable comments from those who read preliminary parts of this thesis. I would like to thank Helene Wolleb, Dennis Komm, Claudia Graf and Stefan Willi. I am grateful to my parents who supported me throughout my studies. ii Contents Contents iii 1 Introduction1 1.1 Motivation.............................. 1 1.2 Overview............................... 3 1.3 Contributions ............................ 3 2 Request Handling5 2.1 Storage System Model....................... 5 2.2 Arrival Only – the Early Morning Scenario........... 7 2.2.1 The Always-Return-to-the-Door Robot......... 7 2.2.2 The Direct Robot...................... 9 2.2.3 The Box-Swapping Robot................. 12 2.2.4 Online Request Arrival .................. 13 2.3 Arrival Only with Two Doors................... 15 2.3.1 Fixed Door Assignment.................. 16 2.3.2 Hardness of Flexible Door Assignment......... 17 2.3.3 Approximation of Flexible Door Assignment . 21 2.4 Arrivals and Departures - the Entire Day............ 30 2.4.1 Lunch Break with a Full Warehouse........... 31 2.4.2 Single Customer per Slot ................. 32 2.4.3 Hardness for Reusable Slots................ 34 2.5 Related Work ............................ 37 2.5.1 Similar Problems on Finding Permutations . 38 2.5.2 Interval Graph Colouring................. 39 3 Rearrangement 41 3.1 Problem Description and Notation................ 42 3.2 Minimizing the Travel Time.................... 44 3.2.1 General Bounds....................... 44 iii Contents 3.2.2 Sorting on a Path...................... 46 3.2.3 Sorting on a Tree...................... 51 3.2.4 Sorting on a Circle..................... 56 3.2.5 Sorting on General Graphs................ 70 3.2.6 Implementation and Visualization............ 71 3.3 Minimizing the Box Handling Time ............... 71 3.3.1 Sorting on a Path...................... 72 3.3.2 Sorting on a Tree...................... 73 3.4 Related Work ............................ 75 3.4.1 Physical Sorting Algorithms ............... 75 3.4.2 Hard Problems for Travelling on Graphs . 77 3.4.3 Croatian Task Kutije.................... 78 3.4.4 Tait’s Counter Puzzle ................... 81 4 Other Bicycle Storage Systems 85 4.1 Cylindrical Storage Systems.................... 85 4.2 Radhaus in Offenburg ....................... 87 4.3 Conveyor Belt Systems....................... 88 5 Conclusion 91 5.1 Summary of Contributions .................... 91 5.2 Future Work............................. 93 Bibliography 95 iv Chapter 1 Introduction 1.1 Motivation Dozens or even hundreds of bicycles chaotically parked around train sta- tions is a common phenomenon. Daily commuters on their way to catch a train want to get rid of their two-wheeler as quickly as possible. Some store it in bike lockers but many also just leave it along or on the pedestrian walk- way, where they are vulnerable to theft and vandalism, and might obstruct passageways. In the last few years, various automated bicycle storage systems were de- signed and developed for this reason. Customers hand their bikes over to a robotic system that stores them at a safe location. The robot is in charge of managing the available storage space and ensures that no one else but the owner can reclaim a bicycle. Once the customer returns to the locker, the robot retrieves the right bicycle as quickly as possible. One such system is called Bike Loft and was developed by Armin Wytten- bach and tested as a prototype in Winterthur in 2014 [60]. In this prototype, a single robot drives along a linear track. Each bicycle is stored in a separate cuboid box so that commuters can also store helmets or rain jackets in the box alongside their bicycle. These boxes are then stored along this track in a two-dimensional grid on both the left and right side of the track. A door is located at one end of the track where customers can interact with the robot. Whenever a customer arrives with his bicycle, the robot brings an empty box to this door. After the customer stores his belongings inside the box, the door closes and the robot carries the box to one of the empty storage slots along the track. When the customer reclaims his bicycle later on, the robot fetches his box from the corresponding storage slot and brings it to the door. An illustration of this system is shown in Figure 1.1. 1 1. Introduction Figure 1.1: Sketch of a Bike Loft system with a single door at the front and two rows of storage slots on both sides of the track. The robot just brought an empty box to the door so that the customer can load it. This thesis aims at the algorithmic scheduling challenges of such a robotic warehousing system. Where should the robot store a filled box? Should it store the box close to the door because it knows that the next customer arrives only twenty seconds later? Or more to the back because it predicts that this box will only be reclaimed late at night? These questions will lead us to a variety of different challenges. If the robot stores a box close to the door, it does not need to move much and is ready quickly to serve the next customer. So why not always use the free spot that is closest to the door? Well, it might be that these close-by slots are needed for customers arriving during rush hour later on and all other boxes should stay out of the way. Another approach might be to quickly drop the box close to the door first and then later during a less busy time of the day pick it up again and put it to some other place along the track. But how should we decide where the box should go and what is the most efficient way to rearrange the boxes? These scheduling and sorting questions are the problems we will investigate in this thesis. In particular, we look at how such a robot can serve the cus- tomers without letting them wait for their bicycles and how it can efficiently rearrange the stored bicycles internally. All of the models that we consider immediately abstract from the fact that we are dealing with bicycles. Therefore many of our results can be applied to any automated storage system, no matter whether it organizes spare parts in a factory, books in a library or cars in a parking garage. 2 1.2. Overview 1.2 Overview We focus on two problems that arise from modelling a robotic warehousing system. The first problem, which we discuss in Chapter2, is to make sure that the robot can serve the customers without letting them wait. We show that the difficulty of this scheduling problem heavily depends on the way we model the robot, the customers and the warehouse. In some cases it is efficiently solvable, in others it will be -hard to decide whether we can NP serve all customers without letting any of them wait.